EM Wave Power and Poynting Vector MCQ Quiz - Objective Question with Answer for EM Wave Power and Poynting Vector - Download Free PDF

Calculation:

As, Poynting vector

S = E × H

Given: energy transport = negative z-direction

Electric field = positive y-direction

(−k̂) = (+ĵ) × [î]

Hence, according to the vector cross product, the magnetic field should be in the positive x-direction.

Calculation:

Given:

Electric field magnitude, |E| = 8 V/m

Speed of light, c = 3 × 10⁸ m/s

We know the relation between the electric and magnetic field in an electromagnetic wave:

E / B = c

Rearranging for B:

B = E / c

= (8) / (3 × 10⁸)

= 266 × 10^-8 T

∴ The magnitude of the magnetic vector is 266 × 10^-8 T.

The Poynting vector is a fundamental concept in electromagnetism, representing the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field.

The formula for the Poynting vector S is given by:S" id="MathJax-Element-49-Frame" role="presentation" style="position: relative;" tabindex="0">SS" id="MathJax-Element-70-Frame" role="presentation" style="position: relative;" tabindex="0">SS<span style="font-family: var(--bs-body-font-family); font-size: var(--bs-body-font-size); font-weight: var(--bs-body-font-weight); text-align: var(--bs-body-text-align);">is given by:</span>

\( \mathbf{S} = \mathbf{E} \times \mathbf{H} \)

where:

• E" id="MathJax-Element-50-Frame" role="presentation" style="position: relative;" tabindex="0">EE" id="MathJax-Element-71-Frame" role="presentation" style="position: relative;" tabindex="0">EE E E is the electric field vector
• H" id="MathJax-Element-51-Frame" role="presentation" style="position: relative;" tabindex="0">HH" id="MathJax-Element-72-Frame" role="presentation" style="position: relative;" tabindex="0">HH H H is the magnetic field vector

Explanation:

The main concept is the relationship between the electric field (E) and the magnetic field (B) in an electromagnetic wave. According to the theory of electromagnetism, the ratio of the electric field magnitude to the magnetic field magnitude in free space is equal to the speed of light (c).

The speed of light (c) in a vacuum is approximately 3 × 10⁸ m/s.

Given that the speed of light in a vacuum is 3 × 10⁸ m/s:

E / B = 3 × 10⁸ m/s

 Calculation:

The power P can be expressed as:
   
  \( P = \int \vec{S} \times d \vec{a} \)
   
   Where:
    S is the Poynting vector, representing the power per unit area carried by an electromagnetic wave.
    da is the differential area element.
Power dissipated through 0.2 m length of the wire will be equal to the integral Poynting vector over the surface integral i.e.,

\(P=I^2R= \int \vec{S} \times d \vec{a} \)

The current in wire will be :

\(I=V/R=5/16 \ A\)

The resistance of wire of length 0.2 m is 

\(R \propto l \\ R=1.6 \times 0.2 =0.32\ \Omega\)

The power will be :

\( \int \vec{S} \times d \vec{a} =(5/16)^2\times 0.32 =0.0312\)
The integral is 0.031.

Concept:

The Poynting vector describes the magnitude and direction of the flow of energy in electromagnetic waves.

mathematically, the Poynting vector states that:

\(P = \vec E \times \vec H\;Watt/{m^2}\)

Calculation:

Given,

\({\rm{\vec E}} = \left| {{\rm{\vec E}}} \right|{a_x}\)

 \(\vec H = \left| {\vec H} \right|{a_y}\)

∴  \(P = \left| {{\rm{\vec E}}} \right|\left| {\vec H} \right|.{\vec a_x} \times {\vec a_y}\)

\(= \left| {{\rm{\vec E}}} \right|\left| {\vec H} \right|{\vec a_z}\)

Poynting vector (S):

It states that the cross product of electric field vector (E) and magnetic field vector (H) at any point is a measure of the rate of flow of electromagnetic energy per unit area at that point that is 

\( \vec S = \vec E \times \vec H\)

S = Poynting vector

E = Electric field and

H = Magnetic field

The Poynting vector describes the magnitude and direction of the flow of energy in electromagnetic waves per unit volume.

Application:

Given,

length = l

radious = r

current = i

Potential = V

Since, electric field (E) is the potential per unit length,

Hence, \(E=\frac{V}{l}\)

For a long straight wire conductor, the magnetic field intensity (H) is given by,

\(H=\frac{i}{2\pi r}\)

Hence, the magnitude of pointing vector (S) will be,

S = EH = \(\frac{V}{l}\times \frac{i}{2\pi r}=\frac{Vi}{2\pi rl}\)

Poynting vector:

It states that the cross product of electric field vector (E) and magnetic field vector (H) at any point is a measure of the rate of flow of electromagnetic energy per unit area at that point that is 

\( \vec S = \vec E \times \vec H\)

Where P = Poynting vector,

E = Electric field and

H = Magnetic field

Or, it can be written as,

\(S = \frac{Power}{Area}\)

The Poynting vector describes the magnitude and direction of the flow of energy in electromagnetic waves.

The unit of the Poynting vector is watt/m2.

Additional Information

1) The Poynting vector (i.e. energy flow per unit area per unit time) for a plane electromagnetic wave is given by

\( \vec S = \frac{1}{\mu_o}(\vec E \times \vec H)\)

\( S = \frac{1}{\mu_o}(E H\, sin\theta )\)

2) From the above, it is clear that E and H are mutually perpendicular and also they are perpendicular to the direction of propagation of the wave.

Thus, the direction of the Poynting vector is along the direction of the propagation of the electromagnetic wave.

The Poynting vector describes the magnitude and direction of the flow of energy in electromagnetic waves per unit volume.

∴ The dimensions of both the Poynting vector and the electromagnetic power density are the same, i.e. M1 L-1 T-2

where M = Mass, L = Length, T = Time

The unit of the Poynting vector is Power density.

Explanation:

• The Poynting vector (S) is defined as the rate of energy per unit area.
• The rate of energy is also known as power.  
• Therefore, S = \( {P\over A}\) ​
• ​where, S = Poynting vector , P = Power and A = Area
• Unit of Power is the watt (W) and the area is m².
• So, the SI unit of S = \({W\over m^2}\)

Important Points

The Poynting vector {S} is also defined as:

S = E × H

where, S = Poynting vector

E = Electric field vector

H = Magnetic field vector

× = Cross product

Concept:

1) The unit vector normal to the surface/plane ax + by = c is given by:

\({\hat a_n} = \frac{{a \cdot {{\hat a}_x} + b \cdot {{\hat a}_y}}}{{\sqrt {{a^2} + {b^2}} }}\) 

Where, a, b, c are constants.

2). For a magnetic field vector given as:

\(\vec H = {H_0}\cos \left( {\omega t - \beta x} \right){\hat a_2}\) 

The direction of propagation â_p = â_x and the direction of magnetic field vector â_H = â_z    

3) The average power of the EM wave is given by:

\({\vec p_{avg}} = \frac{1}{2}\frac{{E_0^2}}{{{\eta _0}}}{\hat a_p} = \frac{1}{2}{\eta _0}H_0^2{\hat a_p}\) 

4) Average power passing through a surface s is given by:

\({p_{avg}} = {\vec p_{avg}} \cdot \left( {A\;{{\hat a}_n}} \right)\) 

Where A is the given area â_n is the unit vector normal to the surface s.

Calculation:

\(\vec H = 0.1\cos \left( {\omega t - \beta x} \right)\) 

\({\vec p_{avg}} = \frac{1}{2}{\eta _0}H_0^2{\hat a_p}\) 

\({\vec p_{avg}} = \frac{1}{2} \times 120\;\pi \times {\left( {0.1} \right)^2}{\hat a_n}\) 

= 0.6 π â_x

Area of the square plate A = 10 cm × 10 cm

A = 0.1 m × 0.1 m = 0.01 m²

Unit vectors normal to the plane x + 2y = 1 is:

\({\hat a_n} = \frac{{1 \cdot {{\hat a}_x} + 2 \cdot {{\hat a}_y}}}{{\sqrt {{1^2} + {2^2}} }} = \frac{{{{\hat a}_x} + 2{{\hat a}_y}}}{{\sqrt 5 }}\) 

\(\vec s = A \cdot {\hat a_n} = 0.01 \times \left( {\frac{{{{\hat a}_x} + 2{{\hat a}_y}}}{{\sqrt 5 }}} \right)\) 

Average power passing through the square plate:

\({p_{avg}} = {\vec p_{avg}} \cdot \vec s\) 

\( = \left( {0.6\pi \;{{\hat a}_x}} \right) \cdot \left( {\frac{{0.01}}{{\sqrt 5 }}} \right)\left( {{{\hat a}_x} + 2{{\hat a}_y}} \right)\) 

\( = 0.6\pi \times \frac{{0.01}}{{\sqrt 5 }}\) 

p_avg = 0.00843 ≈ 8.43 mW

Poynting vector:

It states that the cross product of electric field vector (E) and magnetic field vector (H) at any point is a measure of the rate of flow of electromagnetic energy per unit area at that point that is 

\( \vec S = \vec E \times \vec H\)

S = Poynting vector

E = Electric field and

H = Magnetic field

The Poynting vector describes the magnitude and direction of the flow of energy in electromagnetic waves per unit volume.

\(∯_s Re\left( {\vec P } \right)\cdot \hat n\ d\vec s\),  gives average power and it decreases with the increasing radial distance from the source. 

Concept:

Poynting vector:

It states that the cross product of electric field vector (E) and magnetic field vector (H) at any point is a measure of the rate of flow of electromagnetic energy per unit area at that point that is 

\( \vec P = \vec E \times \vec H\)

Where P = Poynting vector, E = Electric field and H = Magnetic field

The Poynting vector describes the magnitude and direction of the flow of energy in electromagnetic waves.

The unit of the Poynting vector is watt/m2 

The Poynting vector will be defined as the multiplication of the magnitude and the unit direction vector.

Where,

The magnitude of the time-average Poynting vector:

\(\left| {\vec P} \right| = \frac{1}{2}η {H^2}\) [∵ |E|/|H| = η ]

The direction of the Poynting vector is the same as the direction of propagation. 

Given:

|E| = 1 V/m

ϵ_r = 4

Analysis:

\(P = \frac{1}{2}\eta {\left| H \right|^2}\)            ---(1)

\(\frac{{\left| E \right|}}{{\left| H \right|}} = \eta \)

\(\left| H \right| = \frac{1}{{120\pi }}\)

Putting in (1)

\(P = \frac{1}{2}\left( {120\pi } \right)\frac{1}{{120\pi \;\times \;120\pi }}\)

= \(\frac{1}{{240\pi }}~W/{m^2}\)

Time average power density vector is given by

\({P_{avg}} = \frac{{E_o^2}}{{2\eta }}\) where \(\eta = \sqrt {\frac{\mu }{\epsilon}} \)

\(\eta = \sqrt {\frac{{1{\mu _o}}}{{4{\epsilon_o}}}} = \frac{{{\eta _o}}}{2} = \frac{{120\pi }}{2} = 60\pi \)

\({P_{avg}} = \frac{{{{\left( 1 \right)}^2}}}{{2.\left( {60\pi } \right)}} = \frac{1}{{120\pi }}W/{m^2}\)

Concept:

Poynting vector:

• It states that the cross product of electric field vector (E) and magnetic field vector (H) at any point is a measure of the rate of flow of electromagnetic energy per unit area at that point that is 

\( \vec P = \vec E × \vec H\)

Or, it can be written as,

\(⇒ S = \frac{Power}{Area}\)

Where P = Poynting vector, E = Electric field and H = Magnetic field

• The Poynting vector describes the magnitude and direction of the flow of energy in electromagnetic waves.
• The unit of the Poynting vector is watt/m2.

EXPLANATION:

The Poynting vector (i.e. energy flow per unit area per unit time) for a plane electromagnetic wave is given by

\( \vec S = \frac{1}{\mu_o}(\vec E \times \vec H)\)

or, \( S = \frac{1}{\mu_o}(E H\, sin\theta )\)

• From the above, it is clear that E and H are mutually perpendicular and also they are perpendicular to the direction of propagation of the wave.
• Thus, the direction of the Poynting vector is along the direction of the propagation of the electromagnetic wave.

• The Poynting vector gives the instantaneous power density. 
• And average Poynting vector gives the average power crossing a unit area placed perpendicular to the incident wave.

Concept:

A TEM wave traveling in the given direction must satisfy the Poynting theorem, as it describes the magnitude and direction of the flow of energy in electromagnetic waves.

Mathematically, the Poynting vector is the cross-product of the Electric field vector and the magnetic field vector, i.e.

\(P = \vec E × \vec H\;Watt/{m^2}\)

Analysis:

Option 1:

With E = +8 ŷ and H = -4 ẑ

According to Poynting theorem, the direction of propagation will be:

P̅ = 8 ŷ × (-4ẑ)

P̅ = - 32 x̂

Since the direction of propagation given is +x direction, Option 1 is incorrect.

Option 2:

With E = -2ŷ and H = -3ẑ  

P̅ = -2ŷ × (-3ẑ)

P̅ = 6 x̂

Since the direction of propagation given is +x direction, Option 2 is correct.

Option 3:

With E = +2ẑ and H = +2ŷ

P̅ = 2ẑ × (2ŷ)

P̅ = - 4 x̂

Since the direction of propagation given is +x direction, Option 3 is incorrect.

Option 4:

With E = -3ŷ and H = +4ẑ  

P̅ = -3ŷ × (4ẑ)

P̅ = - 12 x̂

Since the direction of propagation given is +x direction, Option 4 is incorrect.

Concept:

The Poynting vector states that:

\(P = \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} \times \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H}\)

For a plane wave, the Magnitude of Power is given by:

\(P= \frac{{{E^2}}}{{2{\eta _0}}}\)

η₀ = Intrinsic impedance of free space.

Also, the Time-Averaged Energy Density is given by:

\(\frac{1}{2}\epsilon .{{\left| E \right|}^{2}}~J/{{m}^{3}}\)

Calculation:

Given P_avg.= 3 W/m²

\({P_{avg}} = 3 = \frac{{{{\left| E \right|}^2}}}{{2{\eta _0}}}\)

\(\Rightarrow {\left| E \right|^2} = 2 \times 120\pi \times 3\)

= 240 × 3π

Average power density \(=\frac{1}{2}\epsilon .{{E}^{2}}\)

\(=\frac{1}{2}\times \frac{1\times 240\times 3\pi }{36\pi \times {{10}^{-9}}}\)